American Institute of Mathematical Sciences

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December  2019, 24(12): 6817-6835. doi: 10.3934/dcdsb.2019168

Flocking of Cucker-Smale model with intrinsic dynamics

Lining Ru 1,2, and  Xiaoping Xue 1,,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2. 

Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, China

* Corresponding author: Xiaoping Xue

Received  September 2018 Revised  February 2019 Published  July 2019

Fund Project: The authors are supported by NSF of China grants 11731010 and 11671109.

In this paper, we consider the flocking problem of modified continu- ous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents' own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant $ L $ of the intrinsic dynamics and exponent $ \beta $ measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large $ L $ or unconditionally for $ \beta>0 $. At last, we provide several numerical simulations to illustrate our theoretical results.

Keywords: Cucker-Smale flocking, intrinsic dynamics, Lipschitz function.
Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 91D30.
Citation: Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168
References:
[1]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[3]

H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[4]

S. Bolouki and R. Malham$\acute{e}$, Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2012. doi: 10.1109/Allerton.2012.6483385.  Google Scholar

[5]

Y. Cao, W. Ren, D. Casbeer and C. Schumacher, Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.  doi: 10.1109/TAC.2015.2479926.  Google Scholar

[6]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[7]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[8]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[9]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[13]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.  doi: 10.1137/100785910.  Google Scholar

[14]

F. Dalmao and E. Mordecki, Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.  doi: 10.1109/TAC.2012.2188440.  Google Scholar

[15]

J.-G. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.   Google Scholar

[16]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

[17]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[20]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[21]

C.-H. Li and S.-Y. Yang, A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[22]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[23]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[24]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[25]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[28]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[29]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[30]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[31]

L. Ru, Z. Li and X. Xue, Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.  doi: 10.1016/j.jfranklin.2014.12.007.  Google Scholar

[32]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. Google Scholar

[34]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.  doi: 10.1109/TCNS.2017.2765824.  Google Scholar

[35]

H. Su, G. Chen, X. Wang and Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.  doi: 10.1016/j.automatica.2010.10.050.  Google Scholar

[36]

Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp. doi: 10.1063/1.4929496.  Google Scholar

[37]

Y. Sun, Y. Wang and D. Zhao, Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.  doi: 10.1016/j.physa.2015.08.005.  Google Scholar

[38]

T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.  Google Scholar

[39]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

T. Vicsek, A. Czir$\acute{o}$k, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[41]

T. Vicsek and A. Zefeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[42]

W. Yu, G. Chen and M. Cao, Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.  doi: 10.1109/TAC.2011.2112477.  Google Scholar

[43]

W. Yu, G. Chen, M. Cao and J. Kurths, Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.   Google Scholar

show all references

References:
[1]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[3]

H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.  Google Scholar

[4]

S. Bolouki and R. Malham$\acute{e}$, Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2012. doi: 10.1109/Allerton.2012.6483385.  Google Scholar

[5]

Y. Cao, W. Ren, D. Casbeer and C. Schumacher, Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.  doi: 10.1109/TAC.2015.2479926.  Google Scholar

[6]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[7]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[8]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[9]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.  doi: 10.1142/S0218202509003851.  Google Scholar

[10]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[13]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.  doi: 10.1137/100785910.  Google Scholar

[14]

F. Dalmao and E. Mordecki, Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.  doi: 10.1109/TAC.2012.2188440.  Google Scholar

[15]

J.-G. Dong, Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.   Google Scholar

[16]

J.-G. Dong and L. Qiu, Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.  doi: 10.1109/TAC.2016.2631608.  Google Scholar

[17]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[20]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[21]

C.-H. Li and S.-Y. Yang, A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.  doi: 10.3934/dcdsb.2016062.  Google Scholar

[22]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.  doi: 10.3934/dcds.2014.34.3683.  Google Scholar

[23]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.  doi: 10.1090/qam/1401.  Google Scholar

[24]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.  doi: 10.1142/S0218202514500043.  Google Scholar

[25]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[26]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[27]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[28]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[29]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[30]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.  Google Scholar

[31]

L. Ru, Z. Li and X. Xue, Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.  doi: 10.1016/j.jfranklin.2014.12.007.  Google Scholar

[32]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. Google Scholar

[34]

C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.  doi: 10.1109/TCNS.2017.2765824.  Google Scholar

[35]

H. Su, G. Chen, X. Wang and Z. Lin, Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.  doi: 10.1016/j.automatica.2010.10.050.  Google Scholar

[36]

Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp. doi: 10.1063/1.4929496.  Google Scholar

[37]

Y. Sun, Y. Wang and D. Zhao, Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.  doi: 10.1016/j.physa.2015.08.005.  Google Scholar

[38]

T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.  doi: 10.1142/S0219530513500255.  Google Scholar

[39]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[40]

T. Vicsek, A. Czir$\acute{o}$k, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[41]

T. Vicsek and A. Zefeiris, Collective motion, Physics Reports, 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.  Google Scholar

[42]

W. Yu, G. Chen and M. Cao, Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.  doi: 10.1109/TAC.2011.2112477.  Google Scholar

[43]

W. Yu, G. Chen, M. Cao and J. Kurths, Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.   Google Scholar

Figure 1.  Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 3] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 3] $
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Figure 2.  Evolutions of $ d_V(t) $ and $ d_X(t) $ with 6 particles: {(a)} $ d_V(t) $ for $ t\in [0, \; 3] $; {(b)} $ d_X(t) $ for $ t\in [0, \; 3] $
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Figure 3.  Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 150] $; {(b)} velocity $ V_2 $ for $ t\in [0, \; 150] $
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Figure 4.  Evolutions of $ d_V[t] $ and $ d_X[t] $ with 6 particles: {(a)} $ d_V[t] $ for $ t\in [0, \; 150] $; {(b)} $ d_X[t] $ for $ t\in [0, \; 150] $
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Figure 5.  Dynamics of velocity $ V_2 $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
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Figure 6.  Evolutions of $ d_V[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
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Figure 7.  Evolutions of $ d_X[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $; {(b)} $ \beta = 0.3 $; {(c)} $ \beta = 0.55 $; {(d)} $ \beta = 2.4 $
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